Thread: Statement about the product AB of two square matrices, true or false?

Is this statement true or false

if false a counterexample is needed

if true then an explanation

The product

AB of two square matrices can only have an inverse if A and B

both have inverses.

What ideas have you had so far?

A matrix specifies a linear transformation. If a matrix is (or is not) invertible, what does that tell you about the rank of the associated linear transformation? Multiplying the matrices corresponds to composing the transformations.

Another way of thinking about it- a matrix is invertible if and only if its determinant is non-zero. And det(AB)= det(A)det(B).